nLab
Sandbox

Lemma

Let

𝒞AΠA ADiscA AΓA𝒟 \mathcal{C} \array{ \overset{ \phantom{A} \Pi \phantom{A} }{\longrightarrow} \\ \overset{ \phantom{A} Disc \phantom{A} }{\hookleftarrow} \\ \overset{ \phantom{A} \Gamma \phantom{A} }{ \longrightarrow } } \mathcal{D}

be an adjoint triple, with DiscDisc a fully faithful functor. Denoting the adjunction units/counits as

A\phantom{A} adjunction A\phantom{A}A\phantom{A} unit A\phantom{A}A\phantom{A} counit A\phantom{A}
A\phantom{A} (ΠDisc)(\Pi \dashv Disc) A\phantom{A}A\phantom{A} η ʃ\eta^{ʃ} A\phantom{A}A\phantom{A} ϵ ʃ\epsilon^{ʃ} A\phantom{A}
A\phantom{A} (DiscΓ)(Disc \dashv \Gamma) A\phantom{A}A\phantom{A} η \eta^\flat A\phantom{A}A\phantom{A} ϵ \epsilon^\flat A\phantom{A}

then the following composites of unit/counit components are equal:

(1)(η ΠX )(Πϵ X )=(Γη X ʃ)(ϵ ΓX ʃ)AAAAAAΠDiscΓX ϵ ΓX ʃ ΓX Πϵ X Γη X ʃ ΠX η ΠX ΓDiscΠX \left( \eta^{\flat}_{\Pi X} \right) \circ \left( \Pi \epsilon^\flat_X \right) \;\;=\;\; \left( \Gamma \eta^{ʃ}_{X} \right) \circ \left( \epsilon^{ʃ}_{\Gamma X} \right) \phantom{AAAAAA} \array{ \Pi Disc \Gamma X &\overset{\epsilon^{ʃ}_{\Gamma X}}{\longrightarrow}& \Gamma X \\ {}^{ \mathllap{ \Pi \epsilon^\flat_X } }\big\downarrow && \big\downarrow^{\mathrlap { \Gamma \eta^{ʃ}_{X} } } \\ \Pi X &\underset{ \eta^\flat_{\Pi X} }{\longrightarrow}& \Gamma Disc \Pi X }
Proof

We claim that the following diagram commutes:

ΓX ϵ ΓX ʃ Γη X ʃ ΠDiscΓX ΓDiscΠX Πϵ X ΠDiscΓη X ʃ η ΓDiscΠX ʃ η ΠX ΠX ΠDiscΓDiscΠX ΠX Πη X ʃ Πϵ DiscΠX ΠDiscη ΠX ϵ ΠX ʃ ΠDiscΠX id ΠDiscΠX ΠDiscΠX \array{ && && \Gamma X \\ && & {}^{ \epsilon^ʃ_{\Gamma X} }\nearrow && \searrow^{\mathrlap{ \Gamma \eta^{ʃ}_X }} \\ && \Pi Disc \Gamma X && && \Gamma Disc \Pi X \\ & {}^{ \Pi \epsilon^\flat_X }\swarrow && \searrow^{ \mathrlap{ \Pi Disc \Gamma \eta^{ʃ}_X } } && {}^{\mathllap{ \eta^{ʃ}_{\Gamma Disc \Pi X} }}\nearrow && \nwarrow^{ \mathrlap{ \eta^{\flat}_{\Pi X} } } \\ \Pi X && && \Pi Disc \Gamma Disc \Pi X && && \Pi X \\ & {}_{\mathllap{ \Pi \eta^{ʃ}_X }}\searrow && \swarrow_{\mathrlap{ \Pi \epsilon^{\flat}_{Disc \Pi X} }} && {}_{\mathllap{ \Pi Disc \eta^\flat_{\Pi X} }}\nwarrow && \nearrow_{\mathrlap{ \epsilon^{ʃ}_{\Pi X} }} \\ && \Pi Disc \Pi X && \underset{id_{\Pi Disc \Pi X}}{\longleftarrow} && \Pi Disc \Pi X }

This commutes, because:

  1. the left square is the image under Π\Pi of naturality for ϵ \epsilon^\flat on η X ʃ\eta^{ʃ}_X;

  2. the top square is naturality for ϵ ʃ\epsilon^{ʃ} on Γη X ʃ\Gamma \eta^{ʃ}_X;

  3. the right square is naturality for ϵ ʃ\epsilon^{ʃ} on η ΠX \eta^{\flat}_{\Pi X};

  4. the bottom commuting triangle is the image under Π\Pi of the zig-zag identity for (DiscΓ)(Disc \dashv \Gamma) on ΠX\Pi X.

Finally, also the total bottom composite is the identity morphism id ΠXid_{\Pi X}, due to the zig-zag identity for (ʃDisc)(ʃ \dashv Disc).

Therefore the total composite from ΠDiscΓXΓDiscΠX\Pi Disc \Gamma X \to \Gamma Disc \Pi X along the bottom part of the diagram equals the left hand side of (1), while the composite along the top part of the diagram clearly equals the right hand side of (1).

Proposition

(points-to-pieces transform)

Consider an adjoint quadruple of the form

ΠDiscΓcoDisc:HAAΠAA AADiscAA AAAΓAAA AAcoDiscAAB \Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} \Pi \phantom{AA} }{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } \mathbf{B}

(for instance a cohesive topos over some base topos B\mathbf{B}).

Then for all XXX \in \mathbf{X} the following two natural transformations, constructed from the adjunction units/counits and their inverse morphisms (using idempotency), are equal:

(2)ptp B(Πϵ X )(η ΓX ʃ) 1=(η ΠX ) 1(Γη X ʃ)AAAAAAAΓX Γη X ʃ ΓDiscΠX (η ΓX ʃ) 1 ptp B (η ΠX ) 1 ΠDiscΓX Πϵ X ΠX ptp_{\mathbf{B}} \;\;\coloneqq\;\; \left( \Pi \epsilon^\flat_X \right) \circ \left( \eta^{ʃ}_{\Gamma X} \right)^{-1} \;\;=\;\; \left( \eta^\flat_{\Pi X} \right)^{-1} \circ \left( \Gamma \eta^{ʃ}_X \right) \phantom{AAAAAAA} \array{ \Gamma X & \overset{ \Gamma \eta^{ʃ}_X }{\longrightarrow} & \Gamma Disc \Pi X \\ {}^{ \mathllap{ \left( \eta^{ʃ}_{\Gamma X} \right)^{-1} } }\big\downarrow & \searrow^{ \mathrlap{ ptp_{\mathbf{B}} } } & \big\downarrow^{ \mathrlap{ \left( \eta^\flat_{\Pi X} \right)^{-1} } } \\ \Pi Disc \Gamma X &\underset{ \Pi \epsilon^\flat_X }{\longrightarrow}& \Pi X }

Moreover, the image of these morphisms under DiscDisc equals the following composite:

(3)ptp H:XAϵ X AXAη X ʃAʃX, ptp_{\mathbf{H}} \;\colon\; \flat X \overset{ \phantom{A} \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} X \overset{ \phantom{A} \eta^{ʃ}_X \phantom{A} }{\longrightarrow} ʃ X \,,

hence

(4)ptp H=Disc(ptp B). ptp_{\mathbf{H}} \;=\; Disc(ptp_{\mathbf{B}}) \,.

Either of these morphisms we call the points-to-pieces transform.

Proof

For the second statement, notice that the (DiscΓ)(Disc \dashv \Gamma)-adjunct of

ptp H:DiscΓXAϵ X AXAη X ʃADiscΠX ptp_{\mathbf{H}} \;\colon\; Disc \Gamma X \overset{ \phantom{A} \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} X \overset{ \phantom{A} \eta^{ʃ}_X \phantom{A} }{\longrightarrow} Disc \Pi X

is

(5)ptp H˜=ΓXisoAη ΓX AΓDiscΓXisoAΓϵ X AΓX=id ΓXAΓη X ʃAΓDiscΠX, \widetilde{ ptp_{\mathbf{H}} } \;\;=\;\; \underset{ = id_{\Gamma X} }{ \underbrace{ \Gamma X \underoverset{iso}{ \phantom{A} \eta^{\flat}_{\Gamma X} \phantom{A} }{ \longrightarrow } \Gamma Disc \Gamma X \underoverset{iso}{ \phantom{A} \Gamma \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} \Gamma X }} \overset{ \phantom{A} \Gamma \eta^{ʃ}_X \phantom{A} }{\longrightarrow} \Gamma Disc \Pi X \,,

where under the braces we uses the zig-zag identity.

(As a side remark, for later usage, we observe that the morphisms on the left in (5) are isomorphisms, as shown, by idempotency of the adjunctions.)

From this we obtain the following commuting diagram:

DiscΓX ADiscΓη X ʃA DiscΓDiscΠX isoADisc(η ΠX ) 1A DiscΠX ptp H ϵ DiscΠX id ΠX DiscΠX \array{ Disc \Gamma X &\overset{ \phantom{A} Disc \Gamma \eta^{ʃ}_X \phantom{A} }{\longrightarrow}& Disc \Gamma Disc \Pi X &\underoverset{iso}{ \phantom{A} Disc \left(\eta^{ \flat }_{\Pi X}\right)^{-1} \phantom{A} }{ \longrightarrow }& Disc \Pi X \\ &{}_{\mathllap{ ptp_{\mathbf{H}} }}\searrow& {}^{ \mathllap{ \epsilon^{\flat}_{Disc \Pi X} } } \big\downarrow^{\mathrlap{\simeq}} & \nearrow_{\mathrlap{id_{\Pi X}}} \\ && Disc \Pi X }

Here:

  1. on the left we identified ptp H˜˜=ptp H\widetilde {\widetilde {ptp_{\mathbf{H}}}} \;=\; ptp_{\mathbf{\mathbf{H}}} by applying the formula for (DiscΓ)(Disc \dashv \Gamma)-adjuncts to ptp H˜=Γη X ʃ\widetilde {ptp_{\mathbf{H}}} = \Gamma \eta^{ʃ}_X (5);

  2. on the right we used the zig-zag identity for (DiscΓ)(Disc \dashv \Gamma).

This proves the second statement.

form the (DiscΓ)(Disc \dashv \Gamma)-adjoint:

ΓXisoAη ΓX AΓDiscΓXisoAΓϵ X AΓX=id ΓXAΓη X ʃAΓDiscΠX \underset{ = id_{\Gamma X} }{ \underbrace{ \Gamma X \underoverset{iso}{ \phantom{A} \eta^{\flat}_{\Gamma X} \phantom{A} }{ \longrightarrow } \Gamma Disc \Gamma X \underoverset{iso}{ \phantom{A} \Gamma \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} \Gamma X }} \overset{ \phantom{A} \Gamma \eta^{ʃ}_X \phantom{A} }{\longrightarrow} \Gamma Disc \Pi X

and postcompose

ΓXAΓη X ʃAΓDiscΠXisoA(η ΠX ) 1AΠX \Gamma X \overset{ \phantom{A} \Gamma \eta^{ʃ}_X \phantom{A} }{\longrightarrow} \Gamma Disc \Pi X \underoverset{iso}{ \phantom{A} \left(\eta^{ \flat }_{\Pi X}\right)^{-1} \phantom{A} }{ \longrightarrow } \Pi X

alternatively, form the (ΠDisc)(\Pi \dashv Disc)-adjoint

ΠDiscΓXAΠϵ X AΠXisoAΠη X ʃAΠDiscΠXisoAϵ ΠX ʃAΠX=id ΠX \Pi Disc \Gamma X \overset{ \phantom{A} \Pi \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} \underset{ = id_{\Pi X} }{ \underbrace{ \Pi X \underoverset{iso}{ \phantom{A} \Pi \eta^{ʃ}_X \phantom{A} }{\longrightarrow} \Pi Disc \Pi X \underoverset{iso}{ \phantom{A} \epsilon^{ʃ}_{\Pi X} \phantom{A}}{\longrightarrow} \Pi X } }

and precompose

ΓXA(η ΓX ʃ) 1AΠDiscΓXAΠϵ X AΠX \Gamma X \overset{ \phantom{A} \left(\eta^{ʃ}_{\Gamma X}\right)^{-1} \phantom{A} }{\longrightarrow} \Pi Disc \Gamma X \overset{ \phantom{A} \Pi \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} \Pi X

and more

ΠDiscΓX AΠϵ X A ΠX ΠDiscΓη X Πη X ΠDiscΓDiscΓX Πϵ DiscΓX ΠDiscΓX \array{ \Pi Disc \Gamma X &\overset{ \phantom{A} \Pi \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} & \Pi X \\ {}^{ \mathllap{ \Pi Disc \Gamma \eta^\flat_X } } \big\downarrow^{ \mathrlap{\simeq} } && {}^{\mathllap{\simeq}}\big\downarrow^{\mathrlap{ \Pi \eta^\flat_X }} \\ \Pi Disc \Gamma Disc \Gamma X & \underset{ \Pi \epsilon^{\flat}_{Disc \Gamma X} }{\longrightarrow}& \Pi Disc \Gamma X }

now consider

DiscXAη DiscX ADiscX Disc X \overset{ \phantom{A} \eta^{\sharp}_{Disc X} \phantom{A} }{\longrightarrow} \sharp Disc X

hence

DiscXAη DiscX AcoDiscΓDiscX Disc X \overset{ \phantom{A} \eta^{\sharp}_{Disc X} \phantom{A} }{\longrightarrow} coDisc \Gamma Disc X

and postcompose

DiscXAη DiscX AcoDiscΓDiscXcoDisc(η X ) 1coDiscX Disc X \overset{ \phantom{A} \eta^{\sharp}_{Disc X} \phantom{A} }{\longrightarrow} coDisc \Gamma Disc X \overset{ coDisc \left( \eta^{\flat}_{X} \right)^{-1} }{ \longrightarrow} coDisc X
Lemma

Consider an adjoint triple

DiscΓcoDisc:HAADiscAA AAAΓAAA AAcoDiscAAB Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow} } \mathbf{B}

Then application of the functor Γ\Gamma on any morphism XfYH\mathbf{X} \overset{f}{\to} \mathbf{Y} \;\;\in \mathbf{H} is equal to the operations of

  1. pre-composition with the (DiscΓ)(Disc \dashv \Gamma)-adjunction counit ϵ X \epsilon^\flat_{\mathbf{X}}, followed by passing to the (DiscΓ)(Disc \dashv \Gamma)-adjunct;

  2. post-composition with the (ΓcoDisc)(\Gamma \dashv coDisc)-adjunction unit η Y \eta^{ \sharp }_{\mathbf{Y}}, followed by passing to the (ΓcoDisc)(\Gamma \dashv coDisc)-adjunct:

(6)Γ X,Y=η Y ()˜=()ϵ X ˜. \Gamma_{\mathbf{X}, \mathbf{Y}} \;=\; \widetilde{\eta^\sharp_{\mathbf{Y}} \circ (-)} \;=\; \widetilde{ (-) \circ \epsilon^\flat_{\mathbf{X}} } \,.
Proof

For the first equality, consider the following naturality square for the adjunction hom-isomorphism (this Def.):

Hom B(ΓY,ΓY) ()˜ Hom H(Y,coDiscΓY) Hom B(Γ(f),ΓY) Hom H(f,coDiscΓY) Hom B(ΓX,ΓY) ()˜ Hom H(X,coDiscΓY)AAAAA{ΓYid ΓYΓY} {Yη Y coDiscΓY} {ΓXΓ(f)ΓY} {Xη Y fcoDiscΓY} \array{ Hom_{\mathbf{B}}( \Gamma \mathbf{Y} , \Gamma \mathbf{Y} ) &\overset{\widetilde {(-)}}{\longrightarrow}& Hom_{\mathbf{H}}( \mathbf{Y}, coDisc \Gamma \mathbf{Y} ) \\ {}^{\mathllap{ Hom_{\mathbf{B}}(\Gamma(f), \Gamma \mathbf{Y}) }} \big\downarrow && \!\!\!\!\! \big\downarrow^{\mathrlap{ Hom_{\mathbf{H}}( f, coDisc \Gamma \mathbf{Y} ) }} \\ Hom_{\mathbf{B}}( \Gamma \mathbf{X}, \Gamma \mathbf{Y} ) &\overset{\widetilde{ (-) }}{\longleftarrow}& Hom_{\mathbf{H}}( \mathbf{X}, coDisc \Gamma \mathbf{Y} ) } \phantom{AAAAA} \array{ \{ \Gamma \mathbf{Y} \overset{id_{\Gamma \mathbf{Y}}}{\to} \Gamma \mathbf{Y}\} &\longrightarrow& \{ \mathbf{Y} \overset{\eta^\sharp_{\mathbf{Y}}}{\to} coDisc \Gamma \mathbf{Y} \} \\ \big\downarrow && \big\downarrow \\ \{ \Gamma \mathbf{X} \overset{\Gamma(f)}{\to} \Gamma \mathbf{Y} \} &\longleftarrow& \{ \mathbf{X} \overset{\eta^\sharp_{\mathbf{Y}} \circ f}{\longrightarrow} coDisc \Gamma \mathbf{Y} \} }

Chasing the identity morphism id ΓYid_{\Gamma \mathbf{Y}} through this diagram, yields the claimed equality, as shown on the right. Here we use that the right adjunct of the identity morphism is the adjunction unit, as shown.

The second equality is fomally dual:

Hom B(ΓX,ΓX) ()˜ Hom H(DiscΓX,X) Hom B(ΓX,Γ(f)) Hom X(DiscΓX,f) Hom B(ΓX,ΓY) ()˜ Hom H(DiscΓX,Y)AAAAA{ΓXid ΓXΓX} {DiscΓXϵ X X} {ΓXΓ(f)Γ(Y)} {DiscΓXfϵ X Y} \array{ Hom_{\mathbf{B}}( \Gamma \mathbf{X}, \Gamma \mathbf{X}) &\overset{\widetilde { (-) }}{\longrightarrow}& Hom_{\mathbf{H}}( Disc \Gamma \mathbf{X} , \mathbf{X}) \\ {}^{\mathllap{ Hom_{\mathbf{B}}( \Gamma \mathbf{X}, \Gamma(f) ) }} \big\downarrow && \big\downarrow^{ \mathrlap{ Hom_{\mathbf{X}}( Disc \Gamma \mathbf{X}, f ) } } \\ Hom_{\mathbf{B}}( \Gamma \mathbf{X}, \Gamma \mathbf{Y} ) &\overset{ \widetilde{ (-) } }{\longleftarrow}& Hom_{\mathbf{H}}( Disc \Gamma \mathbf{X}, \mathbf{Y} ) } \phantom{AAAAA} \array{ \{ \Gamma \mathbf{X} \overset{id_{\Gamma \mathbf{X}}}{\to} \Gamma \mathbf{X} \} &\longrightarrow& \{ Disc \Gamma \mathbf{X} \overset{\epsilon^{\flat}_X}{\to} \mathbf{X} \} \\ \big\downarrow && \big\downarrow \\ \{ \Gamma \mathbf{X} \overset{\Gamma(f)}{\to} \Gamma(\mathbf{Y}) \} &\longleftarrow& \{ Disc \Gamma \mathbf{X} \overset{f\circ \epsilon^\flat_{\mathbf{X}} }{\longrightarrow} \mathbf{Y}\} }
Proposition

Consider an adjoint quadruple of the form

ΠDiscΓcoDisc:HAAΠAA AADiscAA AAAΓAAA AAcoDiscAAB \Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} \Pi \phantom{AA} }{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } \mathbf{B}

(for instance a cohesive topos over some base topos B\mathbf{B}).

Then the following are equivalent:

  1. pieces have points:

    1. as seen in B\mathbf{B}: For every object XXX \in \mathbf{X} the points-to-pieces transform (Prop. 1) in B\mathbf{B} (2) is an epimorphism:
    ptp B:ΓXepiΠX ptp_{\mathbf{B}} \;\colon\; \Gamma X \overset{ epi }{\longrightarrow} \Pi X
    1. equivalently, as seen in H\mathbf{H}: For every object XXX \in \mathbf{X} the points-to-pieces transform (Prop. 1) in H\mathbf{H} (3) is an epimorphism:
    ptp H:XepiʃX ptp_{\mathbf{H}} \;\colon\; \flat X \overset{ epi }{\longrightarrow} ʃ X
  2. discrete objects are concrete: For every object SBS \in \mathbf{B} the discrete object Disc(S)Disc(S) is a concrete object, in that the sharp adjunction counit on Disc(S)Disc(S) is a monomorphism:

    η Disc(S) :DiscSmonoDiscS \eta^\sharp_{Disc(S)} \;\colon\; Disc S \overset{ mono }{\longrightarrow} \sharp Disc S
Proof

First observe the equivalence of the first two statements:

ptp His epiAAAiffAAAptp Btestisepi. ptp_{\mathbf{H}} \;\; \text{is epi} \phantom{AAA} \text{iff} \phantom{AAA} ptp_{\mathbf{B}} \;\; \test{is epi} \,.

In one direction, assume that ptp Bptp_{\mathbf{B}} is an epimorphism. By (4) we have ptp H=Disc(ptp B)ptp_{\mathbf{H}} = Disc(ptp_{\mathbf{B}}), but DiscDisc is a left adjoint and left adjoints preserve monomorphisms.

In the other direction, assume that ptp Hptp_{\mathbf{H}} is an epimorphism. By (2) and (5) we see that ptp Bptp_{\mathbf{B}} is re-obtained from this by applying Γ\Gamma and then composition with isomorphisms. But Γ\Gamma is again a left adjoint, and hence preserves epimorphism, as does composition with isomorphisms.

By applying (2) again, we find in particular that pieces have points is also equivalent to Πϵ DiscS \Pi \epsilon^\flat_{Disc S} being an epimorphism, for all SBS \in \mathbf{B}. But this is equivalent to

Hom B(Πϵ X ,S)=Hom H(ϵ X ,Disc(S)) Hom_{\mathbf{B}}(\Pi \epsilon^\flat_{\mathbf{X}}, S) = Hom_{\mathbf{\mathbf{H}}}(\epsilon^\flat_{\mathbf{X}}, Disc(S))

being a monomorphism for all SS (by adjunction isomorphism and definition of epimorphism).

Now by Lemma 2, this is equivalent to

Hom H(X,η Disc(S) ) Hom_{\mathbf{H}}( \mathbf{X}, \eta^\sharp_{Disc(S)} )

being a monomorphism, which is equivalent to η Disc(S) \eta^\sharp_{Disc(S)} being a monomorphism, hence to discrete objects are concrete.

Last revised on June 23, 2018 at 15:59:31. See the history of this page for a list of all contributions to it.